Nonuniversal Scaling Behavior of Barkhausen Noise

Mechanism of subcritical avalanche propagation in three-dimensional disordered systems

We present a numerical study on necessary conditions for the appearance of infinite avalanche below the critical point in disordered systems that evolve throughout metastable states. The representative of those systems is the nonequilibrium athermal random-field Ising model. We investigate the impact on propagation of infinite avalanche of both the interface of flipped spins at the avalanche's starting point and the number of independent islands of flipped spins in the system at the moment when the avalanche starts. To deduce what effects are originated due to finite system's size, and to distinguish them from the real necessary conditions for the appearance of the infinite avalanche, we examined lattices of different sizes as well as other key parameters for the avalanche propagation.

Effects of external noise on threshold-induced correlations in ferromagnetic systems

In the present paper we investigate the impact of the external noise and detection threshold level on the simulation data for the systems that evolve through metastable states. As a representative model of such systems we chose the nonequilibrium athermal random-field Ising model with two types of the external noise, uniform white noise and Gaussian white noise with various different standard deviations, imposed on the original response signal obtained in model simulations. We applied a wide range of detection threshold levels in analysis of the signal and show how these quantities affect the values of exponent γ_{S/T} (describing the scaling of the average avalanche size with duration), the shift of waiting time between the avalanches, and finally the collapses of the waiting time distributions. The results are obtained via extensive numerical simulations on the equilateral three-dimensional cubic lattices of various sizes and disorders.

Avalanche properties in striplike ferromagnetic systems

We present numerical findings on the behavior of the athermal nonequilibrium random-field Ising model of spins at the thin striplike L_{1}×L_{2}×L_{3} cubic lattices with L_{1}<L_{2}<L_{3}. Changing of system sizes highly influences the evolution and shape of avalanches. The smallest avalanches [classified as three-dimension- (3D) like] are unaffected by the system boundaries, the larger are sandwiched between the top and bottom system faces so are 2D-like, while the largest are extended over the system lateral cross section and propagate along the length L_{3} like in 1D systems. Such a structure of avalanches causes double power-law distributions of their size, duration, and energy with larger effective critical exponent corresponding to 3D-like and smaller to 2D-like avalanches. The distributions scale with thickness L_{1} and are collapsible following the proposed scaling predictions which, together with the distributions' shape, might be important for analysis of the Barkhausen noise experimental data for striplike samples. Finally, the impact of system size on external field that triggers the largest avalanche for a given disorder is presented and discussed.

Critical disorder and critical magnetic field of the nonequilibrium athermal random-field Ising model in thin systems

In the present study of the nonequilibrium athermal random-field Ising model we focus on the behavior of the critical disorder R_{c}(l) and the critical magnetic field H_{c}(l) under different boundary conditions when the system thickness l varies. We propose expressions for R_{c}(l) and H_{c}(l) as well as for the effective critical disorder R_{c}^{eff}(l,L) and effective critical magnetic field H_{c}^{eff}(l,L) playing the role of the effective critical parameters for the L×L×l lattices of finite lateral size L. We support these expressions by the scaling collapses of the magnetization and susceptibility curves obtained in extensive simulations. The collapses are achieved with the two-dimensional (2D) exponents for l below some characteristic value, providing thus a numerical evidence that the thin systems exhibit a 2D-like criticality which should be relevant for the experimental analyses of thin ferromagnetic samples.

The critical Barkhausen avalanches in thin random-field ferromagnets with an open boundary

The interplay between the critical fluctuations and the sample geometry is investigated numerically using thin random-field ferromagnets exhibiting the field-driven magnetisation reversal on the hysteresis loop. The system is studied along the theoretical critical line in the plane of random-field disorder and thickness. The thickness is varied to consider samples of various geometry between a two-dimensional plane and a complete three-dimensional lattice with an open boundary in the direction of the growing thickness. We perform a multi-fractal analysis of the Barkhausen noise signals and scaling of the critical avalanches of the domain wall motion. Our results reveal that, for sufficiently small thickness, the sample geometry profoundly affects the dynamics by modifying the spectral segments that represent small fluctuations and promoting the time-scale dependent multi-fractality. Meanwhile, the avalanche distributions display two distinct power-law regions, in contrast to those in the two-dimensional limit, and the average avalanche shapes are asymmetric. With increasing thickness, the scaling characteristics and the multi-fractal spectrum in thicker samples gradually approach the hysteresis loop criticality in three-dimensional systems. Thin ferromagnetic films are growing in importance technologically, and our results illustrate some new features of the domain wall dynamics induced by magnetisation reversal in these systems.

Mechanisms of self-organized criticality in social processes of knowledge creation

In online social dynamics, a robust scale invariance appears as a key feature of collaborative efforts that lead to new social value. The underlying empirical data thus offers a unique opportunity to study the origin of self-organized criticality (SOC) in social systems. In contrast to physical systems in the laboratory, various human attributes of the actors play an essential role in the process along with the contents (cognitive, emotional) of the communicated artifacts. As a prototypical example, we consider the social endeavor of knowledge creation via Questions and Answers (Q&A). Using a large empirical data set from one of such Q&A sites and theoretical modeling, we reveal fundamental characteristics of SOC by investigating the temporal correlations at all scales and the role of cognitive contents to the avalanches of the knowledge-creation process. Our analysis shows that the universal social dynamics with power-law inhomogeneities of the actions and delay times provides the primary mechanism for self-tuning towards the critical state; it leads to the long-range correlations and the event clustering in response to the external driving by the arrival of new users. In addition, the involved cognitive contents (systematically annotated in the data and observed in the model) exert important constraints that identify unique classes of the knowledge-creation avalanches. Specifically, besides determining a fine structure of the developing knowledge networks, they affect the values of scaling exponents and the geometry of large avalanches and shape the multifractal spectrum. Furthermore, we find that the level of the activity of the communities that share the knowledge correlates with the fluctuations of the innovation rate, implying that the increase of innovation may serve as the active principle of self-organization. To identify relevant parameters and unravel the role of the network evolution underlying the process in the social system under consideration, we compare the social avalanches to the avalanche sequences occurring in the field-driven physical model of disordered solids, where the factors contributing to the collective dynamics are better understood.

The dynamics of meaningful social interactions and the emergence of collective knowledge

Collective knowledge as a social value may arise in cooperation among actors whose individual expertise is limited. The process of knowledge creation requires meaningful, logically coordinated interactions, which represents a challenging problem to physics and social dynamics modeling. By combining two-scale dynamics model with empirical data analysis from a well-known Questions &Answers system Mathematics, we show that this process occurs as a collective phenomenon in an enlarged network (of actors and their artifacts) where the cognitive recognition interactions are properly encoded. The emergent behavior is quantified by the information divergence and innovation advancing of knowledge over time and the signatures of self-organization and knowledge sharing communities. These measures elucidate the impact of each cognitive element and the individual actor's expertise in the collective dynamics. The results are relevant to stochastic processes involving smart components and to collaborative social endeavors, for instance, crowdsourcing scientific knowledge production with online games.

Slip statistics of dislocation avalanches under different loading modes

Slowly compressed microcrystals deform via intermittent slip events, observed as displacement jumps or stress drops. Experiments often use one of two loading modes: an increasing applied stress (stress driven, soft), or a constant strain rate (strain driven, hard). In this work we experimentally test the influence of the deformation loading conditions on the scaling behavior of slip events. It is found that these common deformation modes strongly affect time series properties, but not the scaling behavior of the slip statistics when analyzed with a mean-field model. With increasing plastic strain, the slip events are found to be smaller and more frequent when strain driven, and the slip-size distributions obtained for both drives collapse onto the same scaling function with the same exponents. The experimental results agree with the predictions of the used mean-field model, linking the slip behavior under different loading modes.

Hysteresis in the T=0 random-field Ising model: beyond metastable dynamics

We present a numerical study of the zero-temperature response of the Gaussian random-field Ising model to a slowly varying external field, allowing the system to be trapped in microscopic configurations that are not fully metastable. This modification of the standard single-spin-flip dynamics results in an increase in dissipation (hysteresis) somewhat similar to that observed with a finite driving rate. We then study the distribution of avalanches along the hysteresis loop and perform a finite-size scaling analysis that shows good evidence that the critical exponents associated to the disorder-induced phase transition are not modified.

Magnetization reversal in spin patterns with complex geometry

We study field-driven dynamics of spins with antiferromagnetic interactions along the links of a complex substrate geometry, which is modeled by graphs of a controlled connectivity distribution. The magnetization reversal occurs in avalanches of spin flips, which are pinned by the topological constraints of the underlying graph. The hysteresis loop and avalanche sizes are analyzed and classified in terms of the graph's connectivity and clustering. The results are relevant for magnets with a hierarchical spatial inhomogeneity and for design of nanoscale magnetic devices.

Driving rate effects in avalanche-mediated first-order phase transitions

We study the driving-rate and temperature dependence of the power-law exponents that characterize the avalanche distribution in first-order phase transitions. Measurements of acoustic emission in structural transitions in Cu-Zn-Al and Cu-Al-Ni are presented. We show how the observed behavior emerges within a general framework of competing time scales of avalanche relaxation, driving rate, and thermal fluctuations. We confirm our findings by numerical simulations of a prototype model.

Dimensional crossover and universal roughness distributions in Barkhausen noise

We investigate the dimensional crossover of scaling properties of avalanches (domain-wall jumps) in a single-interface model, used for the description of Barkhausen noise in disordered magnets. By varying the transverse aspect ratio A=L(y)/L(x) of simulated samples, the system dimensionality changes from two to three. We find that perturbing away from d=2 is a relevant field. The exponent tau characterizing the power-law scaling of avalanche distributions varies between 1.06(1) for d=2 and 1.275(15) for d=3, according to a crossover function f(x), x identical with (L-1x)(phi)/A, with phi=0.95(3). We discuss the possible relevance of our results to the interpretation of thin-film measurements of Barkhausen noise. We also study the probability distributions of interface roughness, sampled among successive equilibrium configurations in the Barkhausen noise regime. Attempts to fit our data to the class of universality distributions associated to 1/f(alpha) noise give alpha approximately 1-1.1 for d=2 and 3 (provided that suitable boundary conditions are used in the latter case).

Local mean-field study of capillary condensation in silica aerogels

We apply local mean-field (i.e., density functional) theory to a lattice model of a fluid in contact with a dilute, disordered gel network. The gel structure is described by a diffusion-limited cluster aggregation model. We focus on the influence of porosity on both the hysteretic and the equilibrium behavior of the fluid as one varies the chemical potential at low temperature. We show that the shape of the hysteresis loop changes from smooth to rectangular as the porosity increases and that this change is associated with disorder-induced out-of-equilibrium phase transitions that differ in adsorption and in desorption. Our results provide insight in the behavior of 4He in silica aerogels.

Universality of dynamic scaling for avalanches in disordered Ising systems

Dynamic scaling for driven disordered systems is investigated in some disordered Ising models. Using Monte Carlo simulation, we find that avalanches in both random-field and random-bond Ising models follow dynamic power-law scaling in short times, and the scaling relations are universal for the systems studied. The probability distribution of the dynamic scaling exponent theta is found to have two peaks centered at theta(1) and theta(2). The short-time dynamic exponent theta(1) is invariant and universal for all avalanches while the exponent theta(2) depends on the strength of disorder. The analytical result for the early stage evolution of breakdown process in the random-field Ising model is obtained using mean-field approximation. Short-time dynamic scaling is also confirmed.

Finite driving rates in interface models of Barkhausen noise

We consider a single-interface model for the description of Barkhausen noise in soft ferromagnetic materials. Previously, the model was used only in the adiabatic regime of infinitely slow field ramping. We introduce finite driving rates and analyze the scaling of event sizes and durations for different regimes of the driving rate. Coexistence of intermittency, with nontrivial scaling laws, and finite-velocity interface motion is observed for high enough driving rates. Power spectra show a decay approximately omega(-t), with t<2 for finite driving rates, revealing the influence of the internal structure of avalanches.

Dynamic scaling for avalanches in disordered systems

Dynamic scaling for fracture or breakdown process in disordered systems is investigated in a two-dimensional random field Ising model (RFIM). We find two evolving stages in the avalanche process in the RFIM. At the short-time regime, a power-law growth of the avalanche size Deltas is observed; and at late times, the conventional nucleation and growth process is found. At the critical point of the RFIM, the avalanche size is found to obey the dynamic scaling law Delta(s) approximately equal t((d-beta/nu)/z). From this dynamic scaling relation, the critical strength of the random field D(c) and the critical exponents, beta, nu, and z, are determined. The observed dynamics is explained by a simple nucleation theory of first-order phase transformations.

Experimental observation of disorder-driven hysteresis-loop criticality

We have studied the effect of magnetic disorder on the magnetization reversal process in thin Co/CoO films. The antiferromagnetic CoO layer allows a reversible tuning of the magnetic disorder by simple temperature variation. For temperatures above a critical temperature T(c), we observe a discontinuous magnetization reversal, whereas smooth magnetization loops occur for T<T(c). Our measurements establish the existence of a disorder-driven critical point in the nonequilibrium phase diagram. In addition, we observe scaling behavior in the vicinity of the critical point and determine the critical exponents beta = 0.022+/-0.006 and betadelta = 0.30+/-0.03 for this two-dimensional system.

Studying avalanches in the ground state of the two-dimensional random-field ising model driven by an external field

We study the exact ground state of the two-dimensional random-field Ising model as a function of both the external applied field B and the standard deviation sigma of the Gaussian random-field distribution. The equilibrium evolution of the magnetization consists in a sequence of discrete jumps. These are very similar to the avalanche behavior found in the out-of-equilibrium version of the same model with local relaxation dynamics. We compare the statistical distributions of magnetization jumps and find that both exhibit power-law behavior for the same value of sigma. The corresponding exponents are compared.

Scaling of avalanche queues in directed dissipative sandpiles

Using numerical simulations and analytical methods we study a two-dimensional directed sandpile automaton with nonconservative random defects (concentration c) and varying driving rate r. The automaton is driven only at the top row and driving rate is measured by the number of added particles per time step of avalanche evolution. The probability distribution of duration of elementary avalanches at zero driving rate is exactly given by P1(t,c)=t(-3/2) exp[t ln(1-c)]. For driving rates in the interval 0<r</=1 the avalanches are queuing one after another, increasing the periods of noninterrupted activity of the automaton. Recognizing the probability P1 as a distribution of service time of jobs arriving at a server with frequency r, the model represents an example of the class <E,1,GI/infinity,1> server queue in the queue theory. We study scaling properties of the busy period and dissipated energy of sequences of noninterrupted activity. In the limit c-->0 and varying linear system size L<<1/c we find that at driving rates r</=L(-1/2) the distributions of duration and energy of the avalanche queues are characterized by a multifractal scaling and we determine the corresponding spectral functions. For L>>1/c increasing the driving rate somewhat compensates for the energy losses at defects above the line r approximately sqrt[c]. The scaling exponents of the distributions in this region of phase diagram vary approximately linearly with the driving rate. Using properties of recurrent states and the probability theory we determine analytically the exact upper bound of the probability distribution of busy periods. In the case of conservative dynamics c=0 the probability of a continuous flow increases as F(infinity) approximately r(2) for small driving rates.

Barkhausen avalanches in anisotropic ferromagnets with 180 degrees domain walls

We show that Barkhausen noise in two-dimensional disordered ferromagnets with extended domain walls is characterized by the avalanche size exponent tau(s)=1.54 at low disorder. With increasing disorder the characteristic domain size is reduced relative to the system size due to nucleation of new domains and a dynamic phase transition occurs to the scaling behavior with tau(s)=1.30. The exponents decrease at finite driving rate. The results agree with recently observed behavior in amorphous Metglas and Fe-Co-B ribbons when the applied anisotropic stress is varied.

Dynamics of a domain wall in soft-magnetic materials: barkhausen effect and relation with sandpile models

The CZDE model [P. Cizeau, S Zapperi, G. Durin, and H. E. Stanley, Phys. Rev. Lett. 79, 4669 (1997)] for the dynamics of a domain wall in soft-magnetic materials is investigated. The equation of motion for the domain wall is reduced to a dimensionless form where the control parameters are clearly identified. In this way we show that in soft-magnetic materials with low anisotropies the noise can be approximated by a columnar disorder, and perturbation theory gives a good estimate of the avalanche exponents. Moreover, the resulting exponents are found to be identical to those obtained for directed Abelian sandpile models. The analogies and differences with these models and the question of self-organized criticality in the Barkhausen effect are discussed.

Stress-induced recrystallization of a protein crystal by electron irradiation

Ordering of a system of particles into its thermodynamically stable state usually proceeds by thermally activated mass transport of its constituents. Particularly at low temperature, the activation barrier often hinders equilibration--this is what prevents a glass from crystallizing and a pile of sand from flattening under gravity. But if the driving force for mass transport (that is, the excess energy of the system) is increased, the activation barrier can be overcome and structural changes are initiated. Here we report the reordering of radiation-damaged protein crystals under conditions where transport is initiated by stress rather than by thermal activation. After accumulating a certain density of radiation-induced defects during observation by transmission electron microscopy, the distorted crystal recrystallizes. The reordering is induced by stress caused by the defects at temperatures that are low enough to suppress diffusive mass transport. We propose that this defect-induced reordering might be a general phenomenon.